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# Perspective-n-Point (PnP) pose computation {#calib3d_solvePnP} |
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## Pose computation overview |
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The pose computation problem @cite Marchand16 consists in solving for the rotation and translation that minimizes the reprojection error from 3D-2D point correspondences. |
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The `solvePnP` and related functions estimate the object pose given a set of object points, their corresponding image projections, as well as the camera intrinsic matrix and the distortion coefficients, see the figure below (more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward and the Z-axis forward). |
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 |
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Points expressed in the world frame \f$ \bf{X}_w \f$ are projected into the image plane \f$ \left[ u, v \right] \f$ |
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using the perspective projection model \f$ \Pi \f$ and the camera intrinsic parameters matrix \f$ \bf{A} \f$ (also denoted \f$ \bf{K} \f$ in the literature): |
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\f[ |
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\begin{align*} |
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\begin{bmatrix} |
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u \\ |
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v \\ |
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1 |
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\end{bmatrix} &= |
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\bf{A} \hspace{0.1em} \Pi \hspace{0.2em} ^{c}\bf{T}_w |
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\begin{bmatrix} |
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X_{w} \\ |
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Y_{w} \\ |
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Z_{w} \\ |
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1 |
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\end{bmatrix} \\ |
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\begin{bmatrix} |
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u \\ |
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v \\ |
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1 |
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\end{bmatrix} &= |
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\begin{bmatrix} |
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f_x & 0 & c_x \\ |
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0 & f_y & c_y \\ |
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0 & 0 & 1 |
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\end{bmatrix} |
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\begin{bmatrix} |
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1 & 0 & 0 & 0 \\ |
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0 & 1 & 0 & 0 \\ |
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0 & 0 & 1 & 0 |
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\end{bmatrix} |
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\begin{bmatrix} |
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r_{11} & r_{12} & r_{13} & t_x \\ |
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r_{21} & r_{22} & r_{23} & t_y \\ |
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r_{31} & r_{32} & r_{33} & t_z \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix} |
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\begin{bmatrix} |
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X_{w} \\ |
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Y_{w} \\ |
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Z_{w} \\ |
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1 |
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\end{bmatrix} |
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\end{align*} |
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\f] |
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The estimated pose is thus the rotation (`rvec`) and the translation (`tvec`) vectors that allow transforming |
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a 3D point expressed in the world frame into the camera frame: |
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\f[ |
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\begin{align*} |
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\begin{bmatrix} |
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X_c \\ |
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Y_c \\ |
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Z_c \\ |
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1 |
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\end{bmatrix} &= |
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\hspace{0.2em} ^{c}\bf{T}_w |
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\begin{bmatrix} |
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X_{w} \\ |
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Y_{w} \\ |
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Z_{w} \\ |
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1 |
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\end{bmatrix} \\ |
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\begin{bmatrix} |
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X_c \\ |
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Y_c \\ |
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Z_c \\ |
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1 |
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\end{bmatrix} &= |
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\begin{bmatrix} |
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r_{11} & r_{12} & r_{13} & t_x \\ |
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r_{21} & r_{22} & r_{23} & t_y \\ |
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r_{31} & r_{32} & r_{33} & t_z \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix} |
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\begin{bmatrix} |
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X_{w} \\ |
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Y_{w} \\ |
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Z_{w} \\ |
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1 |
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\end{bmatrix} |
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\end{align*} |
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\f] |
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## Pose computation methods |
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@anchor calib3d_solvePnP_flags |
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Refer to the cv::SolvePnPMethod enum documentation for the list of possible values. Some details about each method are described below: |
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- cv::SOLVEPNP_ITERATIVE Iterative method is based on a Levenberg-Marquardt optimization. In |
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this case the function finds such a pose that minimizes reprojection error, that is the sum |
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of squared distances between the observed projections "imagePoints" and the projected (using |
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cv::projectPoints ) "objectPoints". Initial solution for non-planar "objectPoints" needs at least 6 points and uses the DLT algorithm. |
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Initial solution for planar "objectPoints" needs at least 4 points and uses pose from homography decomposition. |
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- cv::SOLVEPNP_P3P Method is based on the paper of X.S. Gao, X.-R. Hou, J. Tang, H.-F. Chang |
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"Complete Solution Classification for the Perspective-Three-Point Problem" (@cite gao2003complete). |
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In this case the function requires exactly four object and image points. |
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- cv::SOLVEPNP_AP3P Method is based on the paper of T. Ke, S. Roumeliotis |
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"An Efficient Algebraic Solution to the Perspective-Three-Point Problem" (@cite Ke17). |
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In this case the function requires exactly four object and image points. |
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- cv::SOLVEPNP_EPNP Method has been introduced by F. Moreno-Noguer, V. Lepetit and P. Fua in the |
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paper "EPnP: Efficient Perspective-n-Point Camera Pose Estimation" (@cite lepetit2009epnp). |
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- cv::SOLVEPNP_DLS **Broken implementation. Using this flag will fallback to EPnP.** \n |
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Method is based on the paper of J. Hesch and S. Roumeliotis. |
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"A Direct Least-Squares (DLS) Method for PnP" (@cite hesch2011direct). |
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- cv::SOLVEPNP_UPNP **Broken implementation. Using this flag will fallback to EPnP.** \n |
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Method is based on the paper of A. Penate-Sanchez, J. Andrade-Cetto, |
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F. Moreno-Noguer. "Exhaustive Linearization for Robust Camera Pose and Focal Length |
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Estimation" (@cite penate2013exhaustive). In this case the function also estimates the parameters \f$f_x\f$ and \f$f_y\f$ |
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assuming that both have the same value. Then the cameraMatrix is updated with the estimated |
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focal length. |
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- cv::SOLVEPNP_IPPE Method is based on the paper of T. Collins and A. Bartoli. |
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"Infinitesimal Plane-Based Pose Estimation" (@cite Collins14). This method requires coplanar object points. |
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- cv::SOLVEPNP_IPPE_SQUARE Method is based on the paper of Toby Collins and Adrien Bartoli. |
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"Infinitesimal Plane-Based Pose Estimation" (@cite Collins14). This method is suitable for marker pose estimation. |
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It requires 4 coplanar object points defined in the following order: |
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- point 0: [-squareLength / 2, squareLength / 2, 0] |
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- point 1: [ squareLength / 2, squareLength / 2, 0] |
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- point 2: [ squareLength / 2, -squareLength / 2, 0] |
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- point 3: [-squareLength / 2, -squareLength / 2, 0] |
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- cv::SOLVEPNP_SQPNP Method is based on the paper "A Consistently Fast and Globally Optimal Solution to the |
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Perspective-n-Point Problem" by G. Terzakis and M.Lourakis (@cite Terzakis2020SQPnP). It requires 3 or more points. |
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## P3P |
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The cv::solveP3P() computes an object pose from exactly 3 3D-2D point correspondences. A P3P problem has up to 4 solutions. |
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@note The solutions are sorted by reprojection errors (lowest to highest). |
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## PnP |
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The cv::solvePnP() returns the rotation and the translation vectors that transform a 3D point expressed in the object |
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coordinate frame to the camera coordinate frame, using different methods: |
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- P3P methods (cv::SOLVEPNP_P3P, cv::SOLVEPNP_AP3P): need 4 input points to return a unique solution. |
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- cv::SOLVEPNP_IPPE Input points must be >= 4 and object points must be coplanar. |
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- cv::SOLVEPNP_IPPE_SQUARE Special case suitable for marker pose estimation. |
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Number of input points must be 4. Object points must be defined in the following order: |
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- point 0: [-squareLength / 2, squareLength / 2, 0] |
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- point 1: [ squareLength / 2, squareLength / 2, 0] |
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- point 2: [ squareLength / 2, -squareLength / 2, 0] |
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- point 3: [-squareLength / 2, -squareLength / 2, 0] |
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- for all the other flags, number of input points must be >= 4 and object points can be in any configuration. |
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## Generic PnP |
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The cv::solvePnPGeneric() allows retrieving all the possible solutions. |
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Currently, only cv::SOLVEPNP_P3P, cv::SOLVEPNP_AP3P, cv::SOLVEPNP_IPPE, cv::SOLVEPNP_IPPE_SQUARE, cv::SOLVEPNP_SQPNP can return multiple solutions. |
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## RANSAC PnP |
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The cv::solvePnPRansac() computes the object pose wrt. the camera frame using a RANSAC scheme to deal with outliers. |
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More information can be found in @cite Zuliani2014RANSACFD |
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## Pose refinement |
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Pose refinement consists in estimating the rotation and translation that minimizes the reprojection error using a non-linear minimization method and starting from an initial estimate of the solution. OpenCV proposes cv::solvePnPRefineLM() and cv::solvePnPRefineVVS() for this problem. |
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cv::solvePnPRefineLM() uses a non-linear Levenberg-Marquardt minimization scheme @cite Madsen04 @cite Eade13 and the current implementation computes the rotation update as a perturbation and not on SO(3). |
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cv::solvePnPRefineVVS() uses a Gauss-Newton non-linear minimization scheme @cite Marchand16 and with an update of the rotation part computed using the exponential map. |
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@note at least three 3D-2D point correspondences are necessary. |
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